P6

Author: allardjun

ProblemSets_PartI/Math227C_P6.tex

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+\documentclass[12pt,letterpaper]{article}
+\usepackage{amsmath,amsthm,amsfonts,amssymb,amscd}
+\usepackage{fullpage}
+\usepackage{lastpage}
+\usepackage{enumerate}
+\usepackage{fancyhdr}
+\usepackage{mathrsfs}
+\usepackage[margin=3cm,bottom=6cm]{geometry}
+\usepackage{wrapfig}
+\usepackage{graphicx}
+
+\setlength{\parindent}{0.0in}
+\setlength{\parskip}{0.05in}
+
+\renewcommand{\theenumi}{\bf\Alph{enumi}}
+
+% Edit these as appropriate
+\newcommand\course{Math 227C}
+\newcommand\semester{Spring 2019}     % <-- current semester
+\newcommand\hwnum{3}                  % <-- homework number
+\newcommand\yourname{Jun Allard} % <-- your name
+%\newcommand\login{jcarberr}           % <-- your CS login
+
+\newenvironment{answer}[1]{
+  \subsubsection*{Problem \hwnum.#1}
+}{\newpage}
+
+\pagestyle{fancyplain}
+\headheight 35pt
+\lhead{ Math 227C}
+\chead{\textbf{ Problem Set 6}}
+%\rhead{Due {\bf Friday, May 11th}}
+\headsep 20pt
+
+\begin{document}
+
+% \begin{enumerate}[A.] % uncomments for multi-problem homeworks
+
+%%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%% PROBLEM %%%%%%%%%%%%%%%%%%
+%\item 
+Many processes, including the spread of an infectious disease through a small community, can be modeled as first-order exponential processes like
+\begin{equation*}
+\frac{dV}{dt} = \frac{1}{\tau}\left(R-1\right) V \quad V(0)=1 \label{eq:exponential}
+\end{equation*}
+where $V$ is the tumor volume, measured in number of cells, and $R$ is a constant.
+
+This will either lead to exponential growth or exponential decay.
+
+The constant $R$ is different for every patient.
+Assume it has Gaussian distribution with mean 1 and standard deviation $\sigma$,
+\begin{equation*}
+p_R(r) = \frac{1}{\sqrt{2\pi \sigma^2}}\, e^{-\left(r-1\right)^2/2\sigma^2}.
+\end{equation*}
+
+\begin{enumerate}[i. ]
+\item Find the probability density function $p_V(v,t)$ of $V(t)$.
+\end{enumerate}
+
+Intuitively, we expect half of the trajectories to grow exponentially, and half of the trajectories to decay exponentially.
+
+\begin{enumerate}[i. ]
+\setcounter{enumi}{1}
+\item Sketch or plot the probability density you found for $p_V(v,t)$.
+\item What is the probability that a trajectory is above the initial condition at $V=1$? In other words, what is $\mathbb{P}(V(t)>1)$? Is it true that half the trajectories remain above the initial condition $V=1$, and half remain below the initial condition $V=1$? 
+\item Suppose $\tau= 1$ months and $\sigma = 0.1$. What percent of patients have a tumor with more than 1000 cells after 10 months?
+\end{enumerate}
+
+A slightly more complicated model that is a modified version of Equation~\ref{eq:exponential}, called the Gompertz model, is used to fit patient data. 
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \end{enumerate} % uncomments for multi-problem homeworks
+\end{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%